## Bayesian inference on sum of random variables

Let $X_1$, $X_2$, ..., $X_n$ be iid RV's with range $[0,1]$ but unknown distribution. (I'm OK with assuming that the distribution is continuous, etc., if necessary.)

Define $S_n = X_1 + \cdots + X_n$.

I am given $S_k$, and ask: What can I infer, in a Bayesian manner, about $S_n$?

That is, I am given the sum of a sample of size $k$ of the RV's, and I would like to know what I can infer about the distribution of the sum of all the RV's, using a Bayesian approach (and assuming reasonable priors about the distribution).

If the range were $\{0,1\}$ instead of $[0,1]$, then this problem is well-studied, and (with uniform priors) you get (scaled) beta distributions for the inferred distribution on $S_n$. But I'm not sure how to approach it with $[0,1]$ as the range...

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 Crossposted here: stats.stackexchange.com/questions/24344 – cardinal Mar 8 2012 at 21:02